Fourier Transforms In Spectroscopy

Preface......Page 5
Contents......Page 7
1.1 Fourier series......Page 11
1.2 Fourier transform......Page 14
1.3 Dirac’s delta function......Page 17
Problems......Page 20
2 General properties of Fourier transforms......Page 23
2.1 Shift theorem......Page 24
2.2 Similarity theorem......Page 25
2.4 Convolution theorem......Page 26
2.5 Power theorem......Page 28
2.7 Derivative theorem......Page 29
2.8 Correlation theorem......Page 30
2.9 Autocorrelation theorem......Page 31
Problems......Page 32
3 Discrete Fourier transform......Page 35
3.1 Effect of truncation......Page 36
3.2 Effect of sampling......Page 39
3.3 Discrete spectrum......Page 43
Problems......Page 47
4.1 Basis of FFT......Page 49
4.2 Cooley–Tukey algorithm......Page 54
4.3 Computation time......Page 56
Problems......Page 58
5.1 Laplace transform......Page 61
5.2 Transfer function of a linear system......Page 66
5.3 z transform......Page 73
Problems......Page 74
6.1 Interference of light......Page 77
6.2 Michelson interferometer......Page 78
6.3 Sampling and truncation in FTS......Page 83
6.5 Apodization......Page 99
6.6 Applications of FTS......Page 100
Problems......Page 106
7.1 Nuclear magnetic moment in a magnetic .eld......Page 109
7.2 Principles of NMR spectroscopy......Page 112
7.3 Applications of NMR spectroscopy......Page 115
8.1 Conventional mass spectrometry......Page 119
8.2 ICR mass spectrometry......Page 121
8.3 Fourier transforms in ICR mass spectrometry......Page 124
9.1 Fraunhofer and Fresnel diffraction......Page 127
9.2 Diffraction through a narrow slit......Page 128
9.3 Diffraction through two slits......Page 130
9.4 Transmission grating......Page 132
9.5 Grating with only three orders......Page 137
9.6 Diffraction through a rectangular aperture......Page 138
9.7 Diffraction through a circular aperture......Page 143
9.8 Diffraction through a lattice......Page 144
10.1 Equivalent width......Page 155
10.2 Moments of a function......Page 158
10.3 Second moment......Page 160
Problems......Page 163
11.1 Interpolation......Page 165
11.2 Mathematical .ltering......Page 170
11.3 Mathematical smoothing......Page 180
11.4 Distortion and (S/N) enhancement in smoothing......Page 184
11.5 Comparison of smoothing functions......Page 190
11.6 Elimination of a background......Page 193
11.7 Elimination of an interference pattern......Page 194
11.8 Deconvolution......Page 196
Problems......Page 200
12.1 Principle of FSD......Page 205
12.2 Signal-to-noise ratio in FSD......Page 212
12.3 Underdeconvolution and overdeconvolution......Page 217
12.4 Band separation......Page 218
12.5 Fourier complex self-deconvolution......Page 219
12.6 Even-order derivatives and FSD......Page 221
Problems......Page 225
13.1 Linear prediction and extrapolation......Page 229
13.2 Extrapolation of linear combinations of waves......Page 230
13.3 Extrapolation of decaying waves......Page 232
13.4 Predictability condition in the spectral domain......Page 233
13.5 Theoretical impulse response......Page 234
13.6 Matrix method impulse responses......Page 236
13.7 Burg’s impulse response......Page 239
13.8 The q-curve......Page 240
13.9 Spectral line narrowing by signal extrapolation......Page 242
13.10 Imperfect impulse response......Page 243
13.11 The LOMEP line narrowing method......Page 248
13.12 Frequency tuning method......Page 250
13.13 Other applications......Page 255
13.14 Summary......Page 258
Problems......Page 259
Answers to problems......Page 261
References......Page 265
Further reading......Page 266
Index......Page 269